Hamiltonian degree sequences in digraphs

We show that for each \eta>0 every digraph G of sufficiently large order n is Hamiltonian if its out- and indegree sequences d^+_1\le ... \le d^+_n and d^- _1 \le ... \le d^-_n satisfy (i) d^+_i \geq i+ \eta n or d^-_{n-i- \eta n} \geq n-i and (ii) d^-_i \geq i+ \eta n or d^+_{n-i- \eta n} \geq n-i for all i < n/2. This gives an approximate solution to a problem of Nash-Williams concerning a digraph analogue of Chv\'atal's theorem. In fact, we prove the stronger result that such digraphs G are pancyclic.Comment: 17 pages, 2 figures. Section added which includes a proof of a conjecture of Thomassen for large tournaments. To appear in JCT

Hamilton decompositions of regular expanders: applications

In a recent paper, we showed that every sufficiently large regular digraph G on n vertices whose degree is linear in n and which is a robust outexpander has a decomposition into edge-disjoint Hamilton cycles. The main consequence of this theorem is that every regular tournament on n vertices can be decomposed into (n-1)/2 edge-disjoint Hamilton cycles, whenever n is sufficiently large. This verified a conjecture of Kelly from 1968. In this paper, we derive a number of further consequences of our result on robust outexpanders, the main ones are the following: (i) an undirected analogue of our result on robust outexpanders; (ii) best possible bounds on the size of an optimal packing of edge-disjoint Hamilton cycles in a graph of minimum degree d for a large range of values for d. (iii) a similar result for digraphs of given minimum semidegree; (iv) an approximate version of a conjecture of Nash-Williams on Hamilton decompositions of dense regular graphs; (v) the observation that dense quasi-random graphs are robust outexpanders; (vi) a verification of the `very dense' case of a conjecture of Frieze and Krivelevich on packing edge-disjoint Hamilton cycles in random graphs; (vii) a proof of a conjecture of Erdos on the size of an optimal packing of edge-disjoint Hamilton cycles in a random tournament.Comment: final version, to appear in J. Combinatorial Theory

Hamilton cycles in dense vertex-transitive graphs

A famous conjecture of Lov\'asz states that every connected vertex-transitive graph contains a Hamilton path. In this article we confirm the conjecture in the case that the graph is dense and sufficiently large. In fact, we show that such graphs contain a Hamilton cycle and moreover we provide a polynomial time algorithm for finding such a cycle.Comment: 26 pages, 3 figures; referees' comments incorporated; accepted for publication in Journal of Combinatorial Theory, series

Hamilton decompositions of regular expanders: a proof of Kelly's conjecture for large tournaments

A long-standing conjecture of Kelly states that every regular tournament on n vertices can be decomposed into (n-1)/2 edge-disjoint Hamilton cycles. We prove this conjecture for large n. In fact, we prove a far more general result, based on our recent concept of robust expansion and a new method for decomposing graphs. We show that every sufficiently large regular digraph G on n vertices whose degree is linear in n and which is a robust outexpander has a decomposition into edge-disjoint Hamilton cycles. This enables us to obtain numerous further results, e.g. as a special case we confirm a conjecture of Erdos on packing Hamilton cycles in random tournaments. As corollaries to the main result, we also obtain several results on packing Hamilton cycles in undirected graphs, giving e.g. the best known result on a conjecture of Nash-Williams. We also apply our result to solve a problem on the domination ratio of the Asymmetric Travelling Salesman problem, which was raised e.g. by Glover and Punnen as well as Alon, Gutin and Krivelevich.Comment: new version includes a standalone version of the `robust decomposition lemma' for application in subsequent paper

Finding Hamilton cycles in robustly expanding digraphs

We provide an NC algorithm for finding Hamilton cycles in directed graphs with a certain robust expansion property. This property captures several known criteria for the existence of Hamilton cycles in terms of the degree sequence and thus we provide algorithmic proofs of (i) an ‘oriented’ analogue of Dirac’s theorem and (ii) an approximate version (for directed graphs) of Chvátal’s theorem